Statistical Topological Data Analysis – A Kernel Perspective

TitleStatistical Topological Data Analysis – A Kernel Perspective
Publication TypeConference Paper
Year of Publication2015
AuthorsKwitt R, Bauer U, Huber S, Niethammer M, Lin W
Conference NameNIPS
Abstract

We consider the problem of statistical computations with persistence diagrams, a summary representation of topological features in data. These diagrams encode persistent homology, a widely used invariant in topological data analysis. While several avenues towards a statistical treatment of the diagrams have been explored recently, we follow an alternative route that is motivated by the success of methods based on the embedding of probability measures into reproducing kernel Hilbert spaces. In fact, a positive definite kernel on persistence diagrams has recently been proposed, connecting persistent homology to popular kernel-based learning techniques such as support vector machines. However, important properties of that kernel which would enable a principled use in the context of probability measure embeddings remain to be explored. Our contribution is to close this gap by proving universality of a variant of the original kernel, and to demonstrate its effective use in two-sample hypothesis testing on synthetic as well as real-world data.

URLhttp://papers.nips.cc/paper/5887-statistical-topological-data-analysis-a-kernel-perspective